Correlation functions and boundary conditions in RCFT and three-dimensional topology

TL;DR

<3-5 sentence high-level summary>This paper constructs correlation functions for rational conformal field theories on possibly non-orientable surfaces with boundary by embedding RCFT data in a 3D topological field theory derived from a modular category. The authors provide a precise map X -> C(X) that lives in $\mathrm{Hom}(W_{\partial X},\mathcal{H}(\hat{X}))$ and prove that these functions satisfy modular invariance and factorization, expressing all structure constants in terms of modular-category data such as fusion (6j) symbols and the S-matrix. Their framework systematizes the computation of correlation functions on general labeled surfaces through connecting 3-manifolds and multiplicity spaces, and they give explicit results for elementary blocks (two- and three-point functions, disk cases, RP^2) and for annulus, Klein bottle, and Möbius strip. The work provides a rigorous 3D-TFT perspective on RCFT correlators, with integrality properties and Verlinde-type relations emerging from the topology, and offers concrete formulas applicable to WZW models via the corresponding modular categories.

Abstract

We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.

Figures (6)

• Figure 1:
• Figure 2:
• Figure 3: The vectors $\psi_i^+$, $\psi_i^-$ are associated to the ribbon graphs in $D^3$ on the left and on the right, respectively
• Figure 4:
• Figure 5: A solid torus with a ribbon graph, whose invariant is the correlation functions of the Möbius strip with boundary condition $a$

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Lemma 3.5
• Proposition 3.6
• Proposition 3.7
• Theorem 3.8